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Field and Galois theory. (English) Zbl 0865.12001

Graduate Texts in Mathematics. 167. New York, NY: Springer. xvi, 281 p. (1996).
This book is written for the beginning graduate student who has a knowledge of groups, rings, and vector spaces. If I were to teach such a course I would definitely use this text. I find the selection of material, the choice of examples, and the writing style to my liking. The material is what one would expect to find and would hope to find in a book of this type. The author has included many interesting examples. The examples illustrate the results and their relationship to other branches of mathematics such as algebraic geometry, number theory, and analysis.
From the author’s preface: There are a number of topics I wanted to have in a single reference source. For instance, most books do not go into the interesting details about discriminants and how to calculate them. There are many versions of discriminants in different fields of algebra. I wanted to address a number of notions of discriminant and give relations between them. For another example, I wanted to discuss both the calculation of the Galois group of a polynomial of degree 3 or 4, which is usually done in Galois theory books, and discuss in detail the calculation of the roots of the polynomial which is usually not done. I feel it is instructive to exhibit the splitting field of a quartic as the top of a tower of simple radical extensions to stress the connection with the solvability of the Galois group. Finally, I wanted a book that does not stop at Galois theory but discusses non-algebraic extensions, especially the extensions that arise in algebraic geometry. The theory of finitely generated extensions makes use of Galois theory and at the same time leads to connections between algebra, analysis, and topology. Such connections are becoming increasingly important in mathematical research, so students should see them early.

MSC:

12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory
12F10 Separable extensions, Galois theory
11R32 Galois theory
12F20 Transcendental field extensions
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
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